Preheat the oven to 350°F. Grease a 9×13-inch baking dish.
Make the sauce: Melt 4 tablespoons butter in a large skillet over medium heat. Add the onion and cook until softened, about 5 to 7 minutes. Stir in the garlic and cook for 30 seconds.
Sprinkle in the flour and cook, stirring constantly, for 1 to 2 minutes.
Slowly whisk in the chicken broth, then the milk until smooth. Stir in the thyme, poultry seasoning, onion powder, garlic powder, and black pepper. Simmer for 8 to 10 minutes, stirring often, until the sauce has thickened.
Remove from the heat and stir in the sour cream. Taste and season with salt as needed.
Fold the shredded chicken into the sauce and spread the mixture evenly into the prepared baking dish.
In a large bowl, combine the stuffing mix, melted butter, and 1¾ cups chicken broth. Stir until evenly moistened and let stand for 5 minutes.
Spoon the stuffing evenly over the chicken mixture without pressing it down.
Bake uncovered for 30 to 35 minutes, or until the filling is bubbling and the stuffing is golden brown.
Let the casserole rest for 5 to 10 minutes before serving. Garnish with chopped parsley, if desired.